They can model how words change in meaning over time [1], they can identify words that mean the same in different languages [2] and use that to translate whole sentences [3] without ever seeing a direct translation.
Of course none of those tasks are solved to perfection yet, but some kind of meaning is definitely being captured.
[1] https://arxiv.org/abs/1703.00607
Also, that's what I mean by "structure" - the context in which tokens are found in text. I don't want to say "syntax" or "semantics" because it seems to me those are loaded terms that have very specific, er, meaning and I am not a linguist to be able to unpick it.
That's why I linked the two papers on unsupervised machine translation. Given enough text in an unknown language, it is already possible to turn those unknown words into words which humans understand.
I don't disagree that humans can extract meaning from language models. But that only speaks to the ability of humans to extract meaning from even very badly formed language. Not to the ability of those models to represent meaning.
Edit: OK, I don't think I understand what you, er, mean. I think you're saying that language models essentially have a kind of understanding of language, and not just the ability to reproduce it without understanding it. Is that what you are saying?